tag:blogger.com,1999:blog-8148573551417578681.post1220780173749850900..comments2020-06-04T20:54:50.955-07:00Comments on Dark Buzz: Copernicanism and many worldsRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-8148573551417578681.post-17855442963748716252012-08-20T21:13:13.889-07:002012-08-20T21:13:13.889-07:00Bob, you express the hopes of most theoretical phy...Bob, you express the hopes of most theoretical physicists. I am in a very small minority. I just think that those hopes are unrealistic, and have led to paradoxes and strange conclusions, such as Aaronson believing in MWI.<br /><br />Yes, categories and other proper classes are formally outside ZFC, but anything proved with classes can also be proved in ZFC. Replace ZFC with some axiomatization that includes classes, if you wish.<br /><br />You can assume the continuum hypothesis if you wish, but I don&#39;t think that it will help you get any closer to a faithful mathematical representation of physical reality.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-35219618506895593282012-08-20T20:34:24.214-07:002012-08-20T20:34:24.214-07:00“[Mathematics] is not an evolving language that we...“[Mathematics] is not an evolving language that we can modify whenever we want.”<br /><br />But we introduce new terms into the subject all the time! In fact, I think that much of the power of mathematics comes from the introduction of new terms. For example, Galois revolutionized mathematics by recognizing the importance of symmetry and formalizing the group concept; Grothendieck did the same when he identified the problems with classical algebraic geometry and introduced the notion of a scheme. Today, some of the biggest research areas (like the study of motives and higher categories) are looking for ways to formalize preexisting intuitions.<br /><br /><br />“[Mathematics] refers to statements that are provable in ZFC.”<br /><br />So category theory is not a part of mathematics? Proper classes are not mathematics? If that’s true, then much of algebra and topology are not mathematics. You might be right that the physical world doesn’t have a faithful representation into ZFC, but I think it’s crazy to say that mathematics consists of only those statements provable in ZFC.<br /><br /><br />“Some intuitions may be illogical and unprovable.”<br /><br />Well, obviously the illogical and contradictory ones are of no interest to mathematicians. But again, I don’t see why a statement has to be provable within ZFC to count as mathematics. Many perfectly legitimate mathematical statements are equivalent to the continuum hypothesis for example. If you want to do mathematics assuming such statements, what’s the problem?<br /><br /><br />“We cannot observe everything.”<br /><br />If something is not observable in principle, then I don’t think it can really be called physical. If we restrict to things that are physical in the sense that they are in principle observable, then I see no reason why we cannot formulate a theory and describe such things mathematically. If the known mathematical concepts do not provide an adequate framework, we can always develop new ones.Bob Jonesnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-11321115989083237452012-08-20T18:25:02.092-07:002012-08-20T18:25:02.092-07:00No, “mathematics” is not just a term that can refe...No, “mathematics” is not just a term that can refer to any abstract language that formalizes our intuitions. It is not an evolving language that we can modify whenever we want. It refers to statements that are provable in <a href="http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory" rel="nofollow">ZFC</a>. Some intuitions may be illogical and unprovable.<br /><br />I agree that it will always be possible to describe our observations mathematically. But It is a huge step from there to say that reality has a faithful representation. We cannot observe everything. We cannot even observe a single atom without irrersibly changing it.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-79141236028166986142012-08-20T13:34:55.225-07:002012-08-20T13:34:55.225-07:00“I do not believe in any of those things, as you c...“I do not believe in any of those things, as you can see from my FQXi essay.”<br /><br />I took a look at your FQXi essay, and I think you’ve got the relationship between math and physics all wrong. You say that there might be no faithful mathematical representation of reality, and you say that scientists are wrong in assuming such a representation exists.<br /><br />In science, we start by observing the world around us, and then we use our observations to formulate theories. A theory, by definition, is something abstracted from the physical world, so it is formulated in an abstract language. The word “mathematics” can refer to any abstract language that formalizes our intuitions, so there’s nothing surprising about the fact that physical theories are formulated using mathematics. It’s just a tautological consequence of the way we do science.<br /><br />Moreover, we can always get a faithful mathematical representation of reality because we’re free to introduce new mathematical notions whenever we like. For example, when physicists first discovered quantum phenomena, they realized that mathematics of classical physics was insufficient to formulate a theory, and this motivated many developments in functional analysis. Quantum particles might seem incomprehensible at first, but if we introduce more abstract terms into the language used to formalize our observations, we get a faithful representation of quantum physics.<br /><br />No matter what new phenomena we discover, it will always be possible to describe our observations mathematically. It doesn’t matter how crazy nature is because mathematics is just a language for expressing our intuitions, and we can always add words to the language. Of course there’s no guarantee that future theories will have the same predictive power that our current ones have (some predictive power was lost, for example, when we went from classical mechanics to quantum mechanics), but at the very least, a theory of physics could just be a record of what nature does at any given moment. At the very least, *that* would be a faithful representation of reality.<br /><br />The problem with your statements is that you misunderstand how scientists use mathematics. In science, mathematics is not a static collection of results that we’re applying to physics; it’s an evolving language that we can modify whenever we want. It’s always possible to construct a faithful representation of reality because we can always introduce new mathematics to describe nature.Bob Jonesnoreply@blogger.com